Functors between Deligne categories and rings of numbers $\sum_{n\in \mathbb Z_+} a_n\binom{s}{n}$ where $s\in \overline{\mathbb Q}$ and $a_n \in \mathbb Z_+$

The semiring $R_+(x)$ is defined as a set of all nonnegative integer linear combinations of binomial coefficients $\binom{x}{n}$ for $n \in \mathbb Z_+$. This paper is concerned with the inquiry into the structure of $R_+(s)$ for complex numbers $s$. Particularly interesting is the case of algebraic $s$ which are not non-negative integers. This question is motivated by the study of functors between Deligne categories $\textrm{Rep}(S_t)$ (and also $\textrm{Rep}(\textrm{GL}_t)$) for $t \in \mathbb C\backslash \mathbb Z_+$. We prove that this semiring is a ring if and only if $s$ is an algebraic number that is not a nonnegative integer. Furthermore, we show that all algebraic integers generated by $s,$ i.e. all elements of $\mathcal O_{\mathbb Q(s)},$ are also contained in this ring. We also give two explicit representations of $R_+(s)$ for both algebraic integers and general algebraic numbers $s.$ One is in terms of inequalities for the valuations with respect to certain prime ideals and the other is in terms of explicitly constructed generators. Moreover, this leads to a particularly simple description of $R_+(s)$ for both quadratic algebraic numbers and roots of unity.